Problem: $\dfrac{ -4r + 9s }{ 8 } = \dfrac{ -7r + 10t }{ -9 }$ Solve for $r$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -4r + 9s }{ {8} } = \dfrac{ -7r + 10t }{ -9 }$ ${8} \cdot \dfrac{ -4r + 9s }{ {8} } = {8} \cdot \dfrac{ -7r + 10t }{ -9 }$ $-4r + 9s = {8} \cdot \dfrac { -7r + 10t }{ -9 }$ Multiply both sides by the right denominator. $-4r + 9s = 8 \cdot \dfrac{ -7r + 10t }{ -{9} }$ $-{9} \cdot \left( -4r + 9s \right) = -{9} \cdot 8 \cdot \dfrac{ -7r + 10t }{ -{9} }$ $-{9} \cdot \left( -4r + 9s \right) = 8 \cdot \left( -7r + 10t \right)$ Distribute both sides $-{9} \cdot \left( -4r + 9s \right) = {8} \cdot \left( -7r + 10t \right)$ ${36}r - {81}s = -{56}r + {80}t$ Combine $r$ terms on the left. ${36r} - 81s = -{56r} + 80t$ ${92r} - 81s = 80t$ Move the $s$ term to the right. $92r - {81s} = 80t$ $92r = 80t + {81s}$ Isolate $r$ by dividing both sides by its coefficient. ${92}r = 80t + 81s$ $r = \dfrac{ 80t + 81s }{ {92} }$